Why an Integrated View Matters
An actuarial dataset can combine exact claim amounts, partial information about losses, incomplete records, and follow-up times that end before every event is observed. The calculations depend on what each record actually tells us. An exact loss contributes a known value to a complete-data summary; a censored loss gives a bound or range, not an exact value; and a missing loss supplies no observed value. For survival follow-up, the risk set at an event time counts subjects still under observation and event-free immediately before that time.
This workshop brings those tasks together without treating partial records as if they were complete. It uses hypothetical data only. No interest rate, mortality assumption, or parametric loss or survival distribution is assumed: the numerical summaries below are empirical calculations from the stated records. A conclusion about a wider population requires additional sampling and observation assumptions.
Start by Classifying What Each Record Says
Before calculating, identify whether an observed amount is exact, censored, or missing. For a right-censored loss recorded at a threshold \(c\), the record says that the loss exceeds \(c\); it does not say that the loss equals \(c\). A missing record gives no observed loss amount or bound. These distinctions determine which calculations can use each row.
For event-time follow-up, a record may instead report an event time or a censoring time. A subject censored at time \(t\) is known to be event-free through the observed follow-up, but the event time after \(t\), if any, is unknown. A risk set at time \(t\) contains the subjects still at risk immediately before \(t\). With tied events and censoring, count all subjects observed up to time \(t\) in the risk set immediately before \(t\); remove the events and censorings at \(t\) when forming the next risk set. This table describes who was at risk; it is not itself a survival estimate.
Worked Example: Classifying Mixed Claim Records
A hypothetical administrator reviews seven records. The reported information is: A, exact loss $80; B, exact loss $120; C, exact loss $120; D, exact loss $200; E, loss greater than $150; F, no loss information; and G, exact loss $300. Classify each record and calculate the complete-case empirical mean and \(\widehat F_n(120)\).
Method. Use only exact losses for these complete-data summaries. Records A, B, C, D, and G are exact, so \(n=5\). Record E is right-censored at $150; it cannot be entered as a $150 loss. Record F is missing and contributes no loss value.
Mean. The sum of exact losses is \(80+120+120+200+300=820\). Therefore,
Empirical distribution at $120. Three of the five exact losses are at most $120: $80, $120, and $120. Thus \(\widehat F_5(120)=3/5=0.6000\).
Check and limitation. The exact values are ordered \(80,120,120,200,300\); their sum is indeed $820, and exactly three are at most $120. These results summarize the five exact records only. They do not establish the mean or distribution of all seven losses, because the censored value is only known to exceed $150 and one value is missing. No interest rate, mortality assumption, or parametric distribution is used. The common mistake avoided is replacing the censored loss by its censoring threshold.
Complete-Data Summaries and Their Scope
For exact observations, \(\widehat F_n(x)\) is the fraction of the included observations no greater than \(x\). Its value changes only at observed losses. The empirical mean is the arithmetic average of those same values. The calculations are direct, but the choice of included records controls the interpretation: if censored or missing records are excluded, the result is a summary of the exact cases, not automatically a summary of the full group.
The empirical distribution formula follows from counting. For each exact value \(x_i\), the indicator \(I(x_i\le x)\) contributes one if that observation falls at or below \(x\), and zero otherwise. Summing the indicators counts the qualifying observations. Dividing by \(n\) converts that count into a proportion, which is precisely the empirical fraction at or below \(x\). This reasoning also explains why inserting an unknown censored value as though it were observed changes the data being summarized.
Worked Example: Comparing Exact-Case Summaries Across Two Groups
Two hypothetical equipment programs report the following loss information, in dollars. Program A has exact losses $40, $80, and $120, plus one loss known to exceed $100. Program B has exact losses $60, $100, $140, and $160, with no additional records. Calculate each program’s complete-case mean and empirical proportion of exact losses at most $100, then state what the comparison does and does not show.
Method. Calculate both summaries using exact observations only, keeping the denominators visible. Program A has three exact observations, and Program B has four.
Program A. The exact-loss sum is \(40+80+120=240\), so its complete-case mean is \(240/3=\$80.00\). Two of the three exact losses are at most $100, so its exact-case empirical proportion at $100 is \(2/3=0.6667\).
Program B. The exact-loss sum is \(60+100+140+160=460\), so its complete-case mean is \(460/4=\$115.00\). Two of the four exact losses are at most $100, giving \(2/4=0.5000\).
Check and limitation. The reported exact-case mean is lower for Program A, while its exact-case proportion at most $100 is higher. The censored Program A loss is greater than $100, but its exact value is unknown; it cannot be added to the mean or treated as an exact observation in the empirical proportion. Even the direction of a comparison of the programs’ full-population means is not determined by these summaries alone. The exact records may not be representative of each program’s losses, particularly if the chance of observing an exact value depends on its size. No interest rate, mortality assumption, or parametric distribution is assumed. The common mistake avoided is comparing complete-case summaries as though every record in each group were observed exactly.
Constructing and Checking Event-Time Risk Sets
A risk-set table is a count summary of follow-up. At each distinct observed event or censoring time \(t\), record \(r_t\), the number at risk immediately before \(t\); \(d_t\), the number of events at \(t\); and \(c_t\), the number censored at \(t\). The update to the next risk set is
The update holds because every subject in the risk set who has an event or is censored at \(t\) leaves observation as an event-free, followed subject after \(t\). The other members remain at risk. In particular, a subject censored at \(t\) belongs to the risk set immediately before \(t\), but not after it. When events and censorings share a time, the risk set before that time includes both groups; separating the event and censor counts makes the update auditable.
Worked Example: Building a Risk Set with Tied Times
In a hypothetical study of eight machine components, observed follow-up times in months are: event at 2; censoring at 3; event at 3; event at 5; censoring at 5; event at 5; censoring at 7; and event at 8. Construct the risk-set table and check its updates.
Method. Begin with all eight components at risk. At each distinct time, count events and censorings at that time, then subtract both counts to obtain the next risk set.
| Time \(t\) | At risk just before \(t\), \(r_t\) | Events \(d_t\) | Censored \(c_t\) | At risk after \(t\) |
|---|---|---|---|---|
| 2 | 8 | 1 | 0 | 7 |
| 3 | 7 | 1 | 1 | 5 |
| 5 | 5 | 2 | 1 | 2 |
| 7 | 2 | 0 | 1 | 1 |
| 8 | 1 | 1 | 0 | 0 |
At time 2, \(8-1-0=7\). At time 3, the risk set is 7, including both the component with an event at 3 and the component censored at 3; the next count is \(7-1-1=5\). At time 5, \(5-2-1=2\). At time 7, \(2-0-1=1\), and at time 8, \(1-1-0=0\).
Check and limitation. The event counts sum to \(1+1+2+0+1=5\), and the censor counts sum to \(0+1+1+1+0=3\), accounting for all eight components. This table gives the observed risk-set counts, not an estimated survival curve or a claim about unobserved follow-up. No interest rate, mortality assumption, or parametric distribution is used. The common mistake avoided is removing a subject censored at a tied time before counting the risk set immediately before that time.
Assumptions and Common Interpretation Traps
Empirical calculations are exact descriptions of the records included, but their relevance to a larger population depends on how the records were collected and observed. For a complete-case loss summary to represent the underlying loss distribution, the exact cases must be suitable for that purpose. If large losses are more likely to be censored, for example, the exact cases can underrepresent large losses. The arithmetic remains correct for the included values; the broader interpretation does not follow automatically.
For a risk-set description, follow-up times and event/censor classifications must be recorded correctly. To interpret a survival analysis beyond these counts, one must also consider whether the censoring process is independent or noninformative for the event time, as discussed in the earlier material on noninformative censoring. A risk-set table alone does not establish that assumption. Nor does it recover an event time that was not observed.
- Do not replace censoring by equality. A loss known to exceed $150 is not an exact loss of $150.
- Do not confuse missing with censored. Censoring supplies a defined partial constraint; missingness may supply no such constraint.
- Report the denominator. A complete-case empirical proportion uses the number of exact observations, which may differ across groups.
- Keep tied-time conventions consistent. Count subjects censored at \(t\) in the risk set immediately before \(t\), then remove them in the update.
- Separate description from inference. A table of observed values does not by itself justify conclusions about unobserved losses or future event times.
Check Your Understanding
For each problem, distinguish what can be calculated directly from what remains unknown. Amounts are hypothetical, and no parametric distribution is assumed.
- A portfolio extract contains exact losses $30, $50, $50, and $90, one loss known to exceed $70, and one missing loss. Find the complete-case mean and \(\widehat F_4(50)\). (Answer: mean $55.00; empirical proportion 0.7500.)
- Six subjects have follow-up records: event at 1, event at 2, censoring at 3, event at 4, censoring at 4, and event at 6. Give the risk set just before each distinct observed time and the count remaining after each time. (Answer: at times 1, 2, 3, 4, 6 the risk sets are 6, 5, 4, 3, 1; after those times the counts are 5, 4, 3, 1, 0.)
- A group has exact losses $10, $20, $40, and $80, plus a loss known to exceed $60. Find the exact-case median, mean, and empirical proportion at most $40. Explain why the censored record is excluded. (Answer: median $30.00; mean $37.50; proportion 0.7500; the censored value is known only to exceed $60, not as an exact amount.)
- At time 5, a risk set contains 9 subjects; 2 have events and 1 is censored at that time. How many remain at risk immediately afterward, and how many were at risk immediately before time 5? (Answer: 6 remain afterward; 9 were at risk immediately before time 5.)
- Program C has exact losses $25, $75, and $100, plus a loss known to exceed $100. Program D has exact losses $50, $80, $120, and $150. Calculate each complete-case mean and state whether those values alone establish which program has the lower population mean. (Answer: Program C mean $66.67; Program D mean $100.00; no, the complete-case summaries alone do not establish which population mean is lower.)